# Oscillation of Mertens’ product formula

Harold G. Diamond[1]; Janos Pintz[2]

• [1] Univ. of Illinois Dept. of Math. 1409 West Green Street Urbana, IL 61801 USA
• [2] Rényi Mathematical Institute Hungarian Academy of Sciences Reáltanoda u. 13-15 Budapest, H-1053, Hungary
• Volume: 21, Issue: 3, page 523-533
• ISSN: 1246-7405

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## Abstract

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Mertens’ product formula asserts that$\prod _{p\le x}\left(1-\frac{1}{p}\right)\phantom{\rule{0.166667em}{0ex}}logx\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{e}^{-\gamma }$as $x\to \infty$. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^{8}$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

## How to cite

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Diamond, Harold G., and Pintz, Janos. "Oscillation of Mertens’ product formula." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 523-533. <http://eudml.org/doc/10897>.

@article{Diamond2009,
abstract = {Mertens’ product formula asserts that$\prod \_\{p \le x\} \Big ( 1 - \frac\{1\}\{p\} \Big )\, \log x \, \rightarrow \, e^\{-\gamma \}$as $x \rightarrow \infty$. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm\{li \} x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.},
affiliation = {Univ. of Illinois Dept. of Math. 1409 West Green Street Urbana, IL 61801 USA; Rényi Mathematical Institute Hungarian Academy of Sciences Reáltanoda u. 13-15 Budapest, H-1053, Hungary},
author = {Diamond, Harold G., Pintz, Janos},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Mertens’ product formula; oscillation; Euler’s constant; Riemann hypothesis; zeta function; Mertens' theorem; oscillation theorem; Tauberian theory},
language = {eng},
number = {3},
pages = {523-533},
publisher = {Université Bordeaux 1},
title = {Oscillation of Mertens’ product formula},
url = {http://eudml.org/doc/10897},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Diamond, Harold G.
AU - Pintz, Janos
TI - Oscillation of Mertens’ product formula
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 523
EP - 533
AB - Mertens’ product formula asserts that$\prod _{p \le x} \Big ( 1 - \frac{1}{p} \Big )\, \log x \, \rightarrow \, e^{-\gamma }$as $x \rightarrow \infty$. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm{li } x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.
LA - eng
KW - Mertens’ product formula; oscillation; Euler’s constant; Riemann hypothesis; zeta function; Mertens' theorem; oscillation theorem; Tauberian theory
UR - http://eudml.org/doc/10897
ER -

## References

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